.TH std::student_t_distribution 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::student_t_distribution \- std::student_t_distribution

.SH Synopsis
   Defined in header <random>
   template< class RealType = double >  \fI(since C++11)\fP
   class student_t_distribution;

   Produces random floating-point values x, distributed according to probability
   density function:

   \\(p(x|n) = \\frac{1}{\\sqrt{n\\pi} } \\cdot
   \\frac{\\Gamma(\\frac{n+1}{2})}{\\Gamma(\\frac{n}{2})} \\cdot
   (1+\\frac{x^2}{n})^{-\\frac{n+1}{2} } \\)p(x|n) =

   1
   √
   nπ

   ·

   Γ(

   n+1
   2

   )
   Γ(

   n
   2

   )

   · ⎛
   ⎜
   ⎝1+

   x2
   n

   ⎞
   ⎟
   ⎠ -

   n+1
   2

   where n is known as the number of degrees of freedom. This distribution is used when
   estimating the mean of an unknown normally distributed value given n + 1 independent
   measurements, each with additive errors of unknown standard deviation, as in
   physical measurements. Or, alternatively, when estimating the unknown mean of a
   normal distribution with unknown standard deviation, given n + 1 samples.

   std::student_t_distribution satisfies all requirements of RandomNumberDistribution.

.SH Template parameters

   RealType - The result type generated by the generator. The effect is undefined if
              this is not one of float, double, or long double.

.SH Member types

   Member type         Definition
   result_type \fI(C++11)\fP RealType
   param_type \fI(C++11)\fP  the type of the parameter set, see RandomNumberDistribution.

.SH Member functions

   constructor   constructs new distribution
   \fI(C++11)\fP       \fI(public member function)\fP
   reset         resets the internal state of the distribution
   \fI(C++11)\fP       \fI(public member function)\fP
.SH Generation
   operator()    generates the next random number in the distribution
   \fI(C++11)\fP       \fI(public member function)\fP
.SH Characteristics
   n             returns the n distribution parameter (degrees of freedom)
                 \fI(public member function)\fP
   param         gets or sets the distribution parameter object
   \fI(C++11)\fP       \fI(public member function)\fP
   min           returns the minimum potentially generated value
   \fI(C++11)\fP       \fI(public member function)\fP
   max           returns the maximum potentially generated value
   \fI(C++11)\fP       \fI(public member function)\fP

.SH Non-member functions

   operator==
   operator!=                compares two distribution objects
   \fI(C++11)\fP                   \fI(function)\fP
   \fI(C++11)\fP(removed in C++20)
   operator<<                performs stream input and output on pseudo-random number
   operator>>                distribution
   \fI(C++11)\fP                   \fI(function template)\fP

.SH Example


// Run this code

 #include <algorithm>
 #include <cmath>
 #include <iomanip>
 #include <iostream>
 #include <map>
 #include <random>
 #include <vector>

 template<int Height = 5, int BarWidth = 1, int Padding = 1, int Offset = 0, class Seq>
 void draw_vbars(Seq&& s, const bool DrawMinMax = true)
 {
     static_assert(0 < Height and 0 < BarWidth and 0 <= Padding and 0 <= Offset);

     auto cout_n = [](auto&& v, int n = 1)
     {
         while (n-- > 0)
             std::cout << v;
     };

     const auto [min, max] = std::minmax_element(std::cbegin(s), std::cend(s));

     std::vector<std::div_t> qr;
     for (typedef decltype(*std::cbegin(s)) V; V e : s)
         qr.push_back(std::div(std::lerp(V(0), 8 * Height,
                                         (e - *min) / (*max - *min)), 8));

     for (auto h{Height}; h-- > 0; cout_n('\\n'))
     {
         cout_n(' ', Offset);

         for (auto dv : qr)
         {
             const auto q{dv.quot}, r{dv.rem};
             unsigned char d[]{0xe2, 0x96, 0x88, 0}; // Full Block: '█'
             q < h ? d[0] = ' ', d[1] = 0 : q == h ? d[2] -= (7 - r) : 0;
             cout_n(d, BarWidth), cout_n(' ', Padding);
         }

         if (DrawMinMax && Height > 1)
             Height - 1 == h ? std::cout << "┬ " << *max:
                           h ? std::cout << "│ "
                             : std::cout << "┴ " << *min;
     }
 }

 int main()
 {
     std::random_device rd{};
     std::mt19937 gen{rd()};

     std::student_t_distribution<> d{10.0f};

     const int norm = 10'000;
     const float cutoff = 0.000'3f;

     std::map<int, int> hist{};
     for (int n = 0; n != norm; ++n)
         ++hist[std::round(d(gen))];

     std::vector<float> bars;
     std::vector<int> indices;
     for (const auto& [n, p] : hist)
         if (float x = p * (1.0f / norm); cutoff < x)
         {
             bars.push_back(x);
             indices.push_back(n);
         }

     for (draw_vbars<8, 5>(bars); const int n : indices)
         std::cout << " " << std::setw(2) << n << "   ";
     std::cout << '\\n';
 }

.SH Possible output:

                         █████                               ┬ 0.3753
                         █████                               │
                   ▁▁▁▁▁ █████                               │
                   █████ █████ ▆▆▆▆▆                         │
                   █████ █████ █████                         │
                   █████ █████ █████                         │
             ▄▄▄▄▄ █████ █████ █████ ▄▄▄▄▄                   │
 ▁▁▁▁▁ ▃▃▃▃▃ █████ █████ █████ █████ █████ ▃▃▃▃▃ ▁▁▁▁▁ ▁▁▁▁▁ ┴ 0.0049
  -4    -3    -2    -1     0     1     2     3     4     5

.SH External links

   Weisstein, Eric W. "Student's t-Distribution." From MathWorld — A Wolfram Web
   Resource.
